TMUA 2021 Paper 1

Two circles have the same radius. The centre of one circle is \((-2, 1)\). The centre of the other circle is \((3, -2)\). The circles intersect at two distinct points. What is the equation of the straight line through the two points at which the circles intersect?

The curve \(y = x^3 - 6x + 3\) has turning points at \(x = \alpha\) and \(x = \beta\), where \(\beta > \alpha\). Find \( \displaystyle\int_{\alpha}^{\beta} x^3 - 6x + 3 \, d x \)

An arithmetic progression and a convergent geometric progression each have first term \(\dfrac{1}{2}\). The sum of the second terms of the two progressions is \(\dfrac{1}{2}\). The sum of the third terms of the two progressions is \(\dfrac{1}{8}\). What is the sum to infinity of the geometric progression?

Find the minimum value of the function \( 2^{2x} - 2^{x+3} + 4 \)

The function \(f\) is such that \( f(m n) = \begin{cases} f(m) f(n) & \quad \text{if } m n \text{is a multiple of 3} \\ m n & \quad \text{if } m n \text{is not a multiple of 3} \end{cases} \) for all positive integers \(m\) and \(n\). Given that \(f(9) + f(16) - f(24) = 0\), what is the value of \(f(3)\)?

The function \(f\) is given by \( f(x) = \dfrac{\cos x + 3}{7 + 5 \cos x - \sin^2 x} \) Find the positive difference between the maximum and the minimum values of \(f(x)\).

The function \(f\) is such that \(f(0) = 0\), and \(x f(x) > 0\) for all non-zero values of \(x\). It is given that \( \displaystyle\int_{-2}^{2} f(x) \, d x = 4 \) and \( \displaystyle\int_{-2}^{2} |f(x)| \, d x = 8 \) Evaluate \( \displaystyle\int_{-2}^{0} f(|x|) \, d x \)

The line \(y = 2x + 3\) meets the curve \(y = x^2 + b x + c\) at exactly one point. The line \(y = 4x - 2\) also meets the curve \(y = x^2 + b x + c\) at exactly one point. What is the value of \(b - c\)?

Find the area enclosed by the graph of \( |x| + |y| = 1 \)

Use the trapezium rule with 3 strips to estimate \( \displaystyle\int_{\dfrac{1}{2}}^{2} 2 \log_10 x \, d x \)

The function \(f\) is given by \( f(x) = x^{\dfrac{1}{7}} (x^2 - x + 1) \) Find the fraction of the interval \(0 < x < 1\) for which \(f(x)\) is decreasing.

The minimum value of the function \(x^4 - p^2 x^2\) is \(-9\). \(p\) is a real number. Find the minimum value of the function \(x^2 - p x + 6\).

The function \(f\) is such that, for every integer \(n\), \( \displaystyle\int_{n}^{n+1} f(x) \, d x = n + 1 \) Evaluate \( \displaystyle\sum_{r=1}^{8} ( \displaystyle\int_{0}^{r} f(x) \, d x ) \)

This question uses radians. Find the number of distinct values of \(x\) that satisfy the equation \( (x+1)(3-x) = 2(1 - \cos(\pi x)) \)

The diagram shows the graph of \(y = f(x)\). The graph consists of alternating straight-line segments of gradient \(1\) and \(-1\) and continues in this way for all values of \(x\). The function \(g\) is defined as \( g(x) = \displaystyle\sum_{r=1}^{10} f(2^{r-1} x) \) Find the value of \( \displaystyle\int_{0}^{1} g(x) \, d x \)

Consider the expansion of \( (a + b x)^n \) The third term, in *ascending* powers of \(x\), is \(105 x^2\). The fourth term, in *ascending* powers of \(x\), is \(210 x^3\). The fourth term, in *descending* powers of \(x\), is \(210 x^3\). Find the value of \(\left(\dfrac{a}{b}\right)^2\).

Which of the following sketches shows the graph of \( \sin(x^2 + y^2) = \dfrac{1}{2} \) where \(x^2 + y^2 \leq 8 \pi\) ?

The curve with equation \( x = y^2 - 6y + 11 \) is rotated \(90^{\circ}\) clockwise about the point \(P\) to give the curve \(C\). \(P\) has \(x\)-coordinate \(-2\) and \(y\)-coordinate \(3\). What is the equation of \(C\)?

The equation \( \sin^2 (4^{\cos \theta} \times 60^{\circ}) = \dfrac{3}{4} \) has exactly three solutions in the range \(0^{\circ} \leq \theta \leq x^{\circ}\). What is the range of all possible values of \(x\)?

Find the length of the curve with equation \( 2 \log_10 (x - y) = \log_10 (2 - 2x) + \log_10 (y + 5) \)